Algorithms to Solve Singularly Perturbed Volterra Integral Equations

Transcription

1 Avalabl a hp://pvamudu/aam Appl Appl Mah ISSN: 9-9 Vol Issu Ju pp 9-8 Prvousl Vol Issu pp Applcaos ad Appld Mahmacs: A Iraoal Joural AAM Algorhms o Solv Sgularl Prurbd Volrra Igral Equaos Marwa Tasr Alqura ad Blal Khar Dparm of Mahmacs ad Sascs Jorda Uvrs of Scc ad Tcholog Irbd Jorda marwa@jusdujo blalhar@ahoocom Rcvd:Spmbr 7 ; Accpd: Fbruar Absrac I hs papr w appl h Dffral Trasform Mhod DTM ad Varaoal Irav Mhod VIM o dvlop algorhms for solvg sgularl prurbd volrra gral quaos SPVIEs Th sud ouls h sgfca faurs of h wo mhods A comparso bw h wo mhods for h soluo of SPVIs s gv for hr ampls Th rsuls show ha boh mhods ar vr ffc cov ad applcabl o a larg class of problms Kwords: Dffral Trasform Mhod Varaoal Irav Mhod Sgularl Prurbd Volrra Igral Equaos MSC No: A B D G Iroduco I rc ars much ao has b pad o fdg soluos for sgularl prurbd volrra gral quaos SPVIEs Th am of hs papr s o cou hs rd ad cosdr 9

2 7 Marwa Tasr Alqura ad Blal Khar w aalcal chqus h Dffral Trasform Mhod DTM ad h Varaoal Irav Mhod VIM for solvg SPVIEs of h form g K d < < whr > s a small posv paramr calld h prurbao paramr ha gvs rs o h sgularl prurbd aur of h problm Alasr Lag ad Smh 988 Agl ad Olmsad 987 Th rl K ad h fuco g ar gv smooh fucos Udr appropra codo o g ad K for vr > has a uqu couous soluo o [] s Brur 98 ad Alasr I should b mod ha ordr o us h DTM h soluo of mus b aalc Th sgularl prurbd aur of arss wh h proprs of h soluo wh > ar compabl wh hos wh For > s a gral quao of h scod d Wh rducs o a gral quao of h frs d whos soluo ma b compabl wh h cas > Problms of hs aur mpl compabl h bhavor of ar Ths suggss h sc of boudar lar ar h org whr h soluo udrgos a rapd raso Brur 98 Agl ad Olmsad 987 Lag ad Smh 988 dvlopd a formal mhodolog o oba asmpoc soluo for Alasr appld a mul-sp mhod o solv sgular prurbao problm Volrra gral quao Fall Alasr ad Moma 8 usd Homoop prurbao mhod o solv h prsd problm I Sco w appl DTM o solv our problm I Sco w us VIM o gv approma soluo for h proposd problm Ts ampls wh ow ac soluos ar prsd a h d of ach sco o dscuss h accurac ad ffcc of h mhods Fall our cocluso wll b gv Sco Solvg SPVIEs Usg DTM Cosdr h gral form of SPVIE whch s gv Now applg dffral rasform o w g H Y G whr H s h dffral rasform of h rl K Thus h rcurrc formula s

3 AAM: Ir J Vol Issu Ju [Prvousl Vol Issu pp ] 7 H Y G Subsug w g g Thrfor h rasformd al codo a s Y g Sarg wh Y ad h rcurrc formula Y ca b drmd Now usg Y Y h Y s asl dfd Coug hs mar h frs N -dffral rasforms of ca b dfd Fall h vrs rasform of Y s Y Dals abou DTM ad s proprs ca b foud Alqura ad Al-Khald Kah ad Arua 9 ad Erur 7 Numrcal Eampls I hs sco w dscuss hr dffr ampls Th rsul wll b compard wh h ac soluo for varous valus of Eampl Cosdr h followg lar problm d whch has h ac soluo Equao ca b wr h form

4 7 Marwa Tasr Alqura ad Blal Khar d Applg DT o w g Y Y 7 Sc h h rasformd al codo s Y Now w codd 7 Mahmaca ad obad Y Y Y Y Y Y 7 Thus h approma soluo aroud ca b prssd as: appr 7 Fgur rprss h absolu rrors bw h ac soluo ad h approma soluo for < < ad cosdrg rms of h DT srs Fgur Absolu rrors of quao usg DTM for N

5 AAM: Ir J Vol Issu Ju [Prvousl Vol Issu pp ] 7 Eampl Cosdr h followg lar problm d 8 h ac soluo s gv b 9 whr ad Equao 9 ca b wr as d d d No ha for d f h s DT s Y F also for g h s DT s Y G Accordgl h dffral rasform for s Y Y { } Y Y

6 7 Marwa Tasr Alqura ad Blal Khar ad h rasformd al codo usd s Y B codg Mahmaca w oba Y Y Y Y Y Th approma soluo s appr Fgur rprss h absolu rrors bw h ac soluo ad h approma soluo for < < ad cosdrg 8 rms of h DT srs Fgur Absolu rrors of Equao 8 usg DTM for N 8 Eampl Cosdr h followg o-lar problm

7 AAM: Ir J Vol Issu Ju [Prvousl Vol Issu pp ] 7 d whch has h ac soluo whr Equao ca b wr h form d L h f F! Also l h h h dffral rasform of h s H Y Y! Applg DT o w g Y G H! sc h al codo s h s rasform Y B usg Y ad h rcurrc formula w oba 8 Y Y Y Y Y Y 7

8 7 Marwa Tasr Alqura ad Blal Khar Thus h approma soluo aroud s appr Th followg abl shows h absolu rror of for dffr valus of Tabl : Th absolu rror of Equao usg DTM for N Solvg SPVIEs Usg VIM Cosdr h gral form of SPVIE gv To appl h VIM o hs problm w hav o dffra o g d g K d < < d To solv w assum ha h rl fuco K s olar wh rspc o Accordg o VIM w ca cosruc h followg corrco fucoal d g K s s ds d d whr s h gral lagrag mulplr whch ca b dfd usg h varaoal hor ad dod h h rao Now mag hs corrco fucoal saoar w oba: d g K s s ds d d

9 AAM: Ir J Vol Issu Ju [Prvousl Vol Issu pp ] 77 d whr g ad d d K s s ds ar rsrcd varaos g ad d K s s ds d Igrag b pars lds d d Thrfor h gral lagrag mulplr sasfs subjc o Solvg h abov quao w g Thus h corrco fucoal bcoms d g K s s ds d d ad h cosdrd al guss s h al codo for h problm g Dals abou h VIM ca b foud H 997 Alawh ad Al-Khald Bazar ad Ghazu 7 Taar ad Dhgha 7 ad Khalgh Gaj ad Sadgh 7

10 78 Marwa Tasr Alqura ad Blal Khar Numrcal Eampls I hs sco w appl h VIM o h sam ampls cosdrd sco ad compar h rsuls wh h ac soluo Eampl Cosdr h problm Eampl ad dffra h quao o g Accordg o VIM h corrco fucoal s d ad h corrco fucoal saoar s d d d Thrfor h gral lagrag mulplr sasfs subjc o Solvg h abov quao lds

11 AAM: Ir J Vol Issu Ju [Prvousl Vol Issu pp ] 79 Thus h corrco fucoal bcoms d 7 Sarg wh h al codo as h al guss Th b 7 whch s h ac soluo Eampl Cosdr h problm Eampl whch ca b wr h form d d d 8 W dffra 8 o g d 9 Th corrco fucoal s d ds s ad h corrco fucoal saoar s d Th obad Lagrag mulplr s

12 8 Marwa Tasr Alqura ad Blal Khar Thus h corrco fucoal bcoms s ds d Sarg wh Th b usg ad wh h hlp of Mahmaca w g rcursvl { / [ 8 8 ]} Coug hs mar s drmd as 8 9 Fgur shows h absolu rror bw h ac soluo ad h approma soluo for < < ad cosdrg s raos of h VIM squc Fgur Absolu rrors of quao 8 usg VIM for N

13 AAM: Ir J Vol Issu Ju [Prvousl Vol Issu pp ] 8 Eampl Cosdr h problm Eampl ad dffra o g d d d Th corrco fucoal s s d s ds d d Now b usg h obad rsul h h gral lagrag mulplr s Thus h corrco fucoal bcoms d s ds d d s Sarg wh h ad h afr Sh[ ] Coug hs mar s drmd as Th followg abl shows h absolu rror of for dffr valus of

14 8 Marwa Tasr Alqura ad Blal Khar Tabl Th absolu rror of quao usg VIM for N Cocluso I hs papr a comparav sud of VIM ad DTM has b coducd Ths mhods wr appld o solv lar ad olar SPVIEs Th hr ampls cosdrd hs wor suppor our blf ha h rsuls of hs mhods ar cll agrm wh ac soluos Th comparso rvald ha alhough h umrcal rsuls ar smlar VIM s much asr mor cov ad mor ffc; dos o rqur rmda compl calculaos such as fdg Talor srs paso volvd h DTM Acowldgm I would l o ha Profssor Haghgh for hs d cooprao I would also l o ha h aomous rfrs for hr -dph radg of ad hr sghful comms o a arlr vrso of hs papr REFERENCES Alawh A Al-Khald K ad Al-Towaq M Rlabl algorhms for solvg gro-dffral quaos wh applcaos Iraoal Joural of Compur Mahmacs Vol 87 No 7 pp 8- Alasr M H Modfd mullag mhod for sgularl prurbd Volrra gral quaos Iraoal Joural of Compur Mahmacs 7 pp - Alasr M H ad Moma S 8 Applcao of homoop prurbao mhod o sgularl prurbd Volrra Igral Equao Joural of Appld Scc 8 pp 7-78 Alqura M ad Al-Khald K Approma Soluos o Nolar Paral Igro- Dffral Equaos wh Applcaos Ha Flow Jorda Joural of Mahmacs ad Sascs JJMS pp 9- Agll J S ad Olmsad W E 987 Sgularl prurbd Volrra gral quaos Sam J Appld Mah 7 pp -

15 AAM: Ir J Vol Issu Ju [Prvousl Vol Issu pp ] 8 Bazar J ad Ghazu H 7 H's varaoal rao mhod for solvg lar ad olar ssms of ordar dffral quaos Appl Mah Compu Vol 9 pp Brur H ad Va Dr Houw 98 Th umrcal soluo of Volrra quaos CWI Amsrdam Erur V S 7 Dffral Trasform Mhod for Solvg Dffral Equaos of La- Emd Tp Mah & Comp Appl Vol No pp -9 H J H 997: A w approach o olar paral dffral quaos Joural of Compu Phscs Vol pp - H J H 997 Varaoal Irao Mhod for Dla Dffral quaos Commu Nolar Sc Vol pp - Kah A ad Arua K 9 Dffral rasform mhod for solvg h lar ad olar Kl Gordo quao Compur Phscs Commucaos 8 pp 78 7 Khalgh H Gaj D ad Sadgh A 7 Applcao of varaoal rao ad homoop-prurbao mhods o olar ha rasfr quaos wh varabl coffcs Numrcal Ha Trasfr Par A - Lag C G ad Smh D R 988 Sgular prurbao aalss of gral quaos Sud Appld Mah 79 pp - Taar M ad Dhgha M 7 O h covrgc of H's varaoal rao mhod Joural Compu Mah Vol 7 No pp -8